Since December 2024, there is a direct non-high-speed train between Paris and Brussels, operated by Ouigo, taking just over 3 hours. There are 3 services every day. The cheapest advertized price is 10€ but you have to book weeks in advance to get it. The maximum price is 59€, which is still cheaper than chaining regional trains.
The existence of this train is somewhat unusual for long distances in France, so I'm leaving the original answer that was valid for the past 10 years below.
Yes, you can travel with regional trains. It takes a little over 4 hours plus the connection times, and there are no good connections, so count on 5 to 6 hours for the journey. The bus takes 4–5 hours and is cheaper than regional trains.
It isn't easy to find the train times or prices because online planners tend to go for the fastest journey only. There's an option to avoid ICE on the German railways site, which avoids other high-speed trains such as Thalys (currently — it didn't in the past). If you can't get a list of local trains connections, look up Bruxelles–Lille, Lille–Amiens and Amiens–Paris separately.
As of today, you can break your journey as follows:
- Paris–Amiens: 21.60€, lasts 66min to 99min, about 1 train per hour.
- Amiens-Lille Flandres: 21.10€, lasts 80min, about 1 train per hour.
- Lille Flandres–Bruxelles with a change at Tournai: 22.60€, lasts 102min, about 1 train per hour.
That's a total of 65.30€. It should be a little less if you go to a train station and buy a single ticket (I don't think you can buy that ticket online).
In comparison, a last minute Thalys ticket costs 99€ and the journey is only 82min. If I elect to travel one day later I find seats for 69€ (obviously this depends on how full the trains are). A last minute ticket on Idbus costs 39€ for a 4-hour journey or 29€ for a 5-hour journey, with no break of journey either way. So regional trains don't seem like a good compromise between price, comfort and speed.